# Why 6.023x10^23 is not written as 6023x10^20?

We know that 1 mole is equal to $6.023\times10^{23}$ atoms. But why it is not written as $6023\times10^{20}$ ?

• How about we write it $602300000000000000000000 \times 10^0$ instead?
– user
Feb 4 '15 at 18:07
• For the same reason for which people write $2.015 \times 10^3$ instead of 2015. :|
– ashu
Feb 4 '15 at 19:13
• @MichaelKjörling how did you count the zeros in what you've written? :D Feb 4 '15 at 20:13

Dissenter's answer is not quite right, since, strictly speaking, as defined there, the scientific notation will work for positive numbers only. Obviously, it can be easily fixed to work for negative as well, but the case of 0 will continue to be problematic. Just try to follow this rule to understand why.

All in all, it is the matter of definitions, but usually, scientific notation is defined as just a way of writing numbers in the form $a \cdot 10^b$ without any restrictions on $a$ which can be any real number. So, both $6.023 \cdot 10^{23}$ and $6023 \cdot 10^{20}$ are considered to be numbers written in scientific notation.

However, in the normalized scientific notation, indeed,

the exponent $b$ is chosen so that the absolute value of $a$ remains at least one but less than ten ($1 ≤ |a| < 10$),

and thus, only $6.023 \cdot 10^{23}$ is written in this notation. Normalized scientific notation is just one special case of scientific notation. And yeah, you could not write down zero in this notation, which is kind of awkward.

Another special case of scientific notation is the so-called engineering notation in which the exponent $b$ is restricted to multiples of 3. Neither $6.023 \cdot 10^{23}$ and $6023 \cdot 10^{20}$ are written in engineering notation, while $602.3 \cdot 10^{21}$ is. Although engineering notation is a special case of scientific one as defined above, is rarely called scientific notation to avoid an ambiguity.

• Looks like Dissenter has updated the answer to account for the comment in your first paragraph, making that obsolete. You may want to revise.
– user
Feb 5 '15 at 8:28

You could write it as either. But the former is scientific notation while the latter is not considered scientific notation.

In scientific notation, you move the decimal place until you have a number [whose absolute value is] between 1 and 10. Then you add a power of ten that tells how many places you moved the decimal.

http://www.factmonster.com/ipka/A0876783.html

To add to all the other answers, in case you're wondering, the reason we like to use the one standard for scientific notation is because it makes comparing sizes very easy. A lot of times in making approximations you only care about the order of magnitude (this is especially true when looking at things like equilibrium constants). Having this standard essentially allows you to compare multiple values/quantities very quickly by disregarding the coefficients and just having a quick glance at their orders of magnitude. If we all used different standards, we'd have to take into account how many digits are before the decimal place.

There are a couple of advantages to using normalized scientific notation (a number less than ten and greater than or equal to one times a power of ten):

• Counting significant figures is easier. All of the digits written in the number before the power of ten are significant when you're using normalized scientific notation.

Writing it other ways isn't incorrect, but it's not as convenient. For example, if you write $1.000\times 10^3$ it's absolutely clear that you've got 4 significant figures, but if you write $1000\times 10^0$ it's not so clear; you can only say "I have at least one significant digit".

• Comparing orders of magnitude is easier. As other answers here note, you don't have to move the decimal point around to compare the powers of ten for a pair of numbers if you're using normalized scientific notation.

If you let the magnitude of the coefficient / significand vary outside the [1,10) interval, then you have to count digits and add that to the exponent to see the magnitude.

With normalized scientific notation, you can see the magnitude at a glance. This is WHY normalized notation is standard practice.